Abstract:

Given a separable strongly self-concordant function f: ℜⁿ → ℜ, we show the associated spectral function F(X)= (f • λ)(X) is also strongly self-concordant function. In addition, there is a universal constant Ο such that, if f(x) is separable self-concordant barrier then Ο²F(X) is a self-concordant barrier. We estimate that for the universal constant we have Ο ≤ 22. This generalizes the relationship between the standard logarithmic barriers -∑_i=1ⁿlog x_i and -log det X and gives a partial solution to a conjecture of L. Tunçel.